Question

\(\int_0^1 (x + 2x + 3x^2 + 4x^3) \, dx \)

• A. 10
• B. \( \frac{5}{2} \)
• C. \( \frac{7}{2} \)
• D. \( \frac{1}{2} \)

Hint:

To simplify the evaluation of integrals, break down the expression into simpler terms, and apply the basic power rule for each term.

Answer 1

The Correct Option is C

We are given: \[ I = \int_0^1 \left( x + 2x + 3x^2 + 4x^3 \right) \, dx \] Simplifying the integrand: \[ I = \int_0^1 (3x + 3x^2 + 4x^3) \, dx \] Now, integrate each term: \[ I = \left[ \frac{3x^2}{2} + \frac{3x^3}{3} + \frac{4x^4}{4} \right]_0^1 \] Evaluating the definite integral: \[ I = \left[ \frac{3(1)^2}{2} + \frac{3(1)^3}{3} + \frac{4(1)^4}{4} \right] - \left[ 0 \right] \] \[ I = \frac{3}{2} + 1 + 1 = \frac{7}{2} \] Thus, the correct answer is \( \frac{7}{2} \).